10
PHYSICAL REVIEWS 8 VOLUME 28, NUMBER 11 Second-quantized theory of Anderson localization in d =2+a 1 DECEMBER 1983 Alba Theumann Instituto de Eisica, Universidade Federal do Bio Grande do Sul, 90000 Porto Alegre, Bio Grande do Su/, Brazil (Received 15 December 1982) Renormalized perturbation-theory methods with dimensional regularization are applied to the lo- calization transition of electrons in random potentials within a field theory where the generating functional is the configurational averaged (Z ) for Z the vacuum-to-vacuum amplitude expressed as a functional integral over Grassman fields, in the replica limit n =0. The bare parameters of the theory are the Fermi level Eo and the variance of the random potential 8'0. Power counting says that at dimensionality d =2+@the theory is super-renormalizable. Renormalization of the inverse propagator by the definition of renormalized Fermi level and interaction, EF and Q = (wR o/Ep )sc, respectively, in order to cancel the single dimensional pole, leads to the same Wilson function P(u) as for the compact nonlinear cr model when the scale parameter ir is varied at constant Wo/Eo. The conductivity is calculated in a perturbation expansion in (~EF), with '=mR'OE+ ' being the inverse lifetime, at d =2+@. It is explicitly shown that in ultraviolet- divergent d-dimensional loop integrals over advanced and retarded propagators, the leading term is regular while the dimensional pole occurs in the next-to-leading term. Then to leading order the conductivity o. o(co) is regular while the first correction that includes the diffusion modes has a di- mensional pole with residue =(im/~ )' . To cancel this pole a renormalized inverse conductance t(u) is defined, and the new Wilson function P(t) obtained by varying x at constant "bare" interac- tion 8'o/EF coincides with the scaling theory of Abrahams, Anderson, Licciardello, and Ramak- rishnan. Scaling laws are derived from the solution of the renormalization-group equation for the conductivity. I. INTRODUCTION The purpose of the present work is to study the locali- zation transition within a field theory where the generat- ing functional Z is the grand partition function for elec- trons in random potentials, or at T=O the vacuum-to- vacuum amplitude, ' by means of renormalized perturba- tion theory. Z is expressed as a functional integral over Grassman (anticommuting) fields. The scaling theory for the localization transition of electrons in random potentials near and above two dimen- sions is now firmly established since Abrahams et al. showed that the dimensionless conductance g at T=O is a universal function of the length scale of the systein. Fol- lowing earlier work by Thouless and Licciardello, they obtained asymptotic forms for the scaling function P(g) to second order in the expansion parameter g ' for dimen- sionality d & 2. Along a different line of approach Wegner first derived scaling laws for the conductivity, while in a series of following papers Wegner and Schaffer ' developed a Lagrangian formulation based on an effective functional that generates the configurational averaged product of two-particle Green's functions (G(zi)G(z2~), „, zi 2 being complex energies. Here the electrons are described by commuting fields and it em- erges from this work that there would be two generating functionals: One for the case sgn(Imzi) =sgn(Imz2) that is mapped into a compact generalized nonlinear o model, and a different functional for sgn(Irnzi)= sgn(lmzz) that maps into a noncompact generalized nonlinear o. models with n =0 components because of the replica method. The difference between the two models is cru- cial: The compact o. model does not have a fixed point for positive physical values of the reduced "temperature" t=g, while the noncompact model exhibits a positive fixed point t~ that describes the mobility edge in an ex- pansion in e=d 2. The conductivity belongs to the last class and the results for the noncompact nonlinear ir model agree with the scaling theory of Abrahams et al. A detailed comparison of both models was given by Houghton et al. Hikami derived an effective Hamiltonian for diffusion modes that also maps into the noncompact nonlinear o model, and he performed a three-loop order calculation in the e expansion. His results show that the second and third loop corrections identically vanish in the replica lim- it n =0, hence confirming the results of Ref. 2 up to O(e'). Efetov et al. were the first to introduce Grassman variables for the electron fields. Also in their work the generating functional is defined to reproduce correctly (G(zi)G(z2) for - sgn(Imzi ) = sgn(Imz2), and it maps into the noncompact nonlinear o. model. Along dif- ferent lines, a self-consistent theory of Anderson localiza- tions was recently developed by Vollhardt and Wolfle. ' The purpose of this paper is twofold. The first purpose is to present a theory based on the fact that for a nonran- dom system the vacuum amplitude Z is the generating functional of all correlation functions at T=O. This is not new, and it has been applied before to the theory of super- conductivity" and the Kondo problem' ' among others. For a randoin system (Z")«=Z„ is the generating func- tional of the configurational averaged correlation func- tions in the n=O limit. The second purpose is to apply 28 6453 1983 The American Physical Society

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Page 1: Second-quantized theory of Anderson localization in

PHYSICAL REVIEWS 8 VOLUME 28, NUMBER 11

Second-quantized theory of Anderson localization in d =2+a

1 DECEMBER 1983

Alba TheumannInstituto de Eisica, Universidade Federal do Bio Grande do Sul, 90000 Porto Alegre, Bio Grande do Su/, Brazil

(Received 15 December 1982)

Renormalized perturbation-theory methods with dimensional regularization are applied to the lo-calization transition of electrons in random potentials within a field theory where the generatingfunctional is the configurational averaged (Z )„for Z the vacuum-to-vacuum amplitude expressedas a functional integral over Grassman fields, in the replica limit n =0. The bare parameters of thetheory are the Fermi level Eo and the variance of the random potential 8'0. Power counting saysthat at dimensionality d =2+@ the theory is super-renormalizable. Renormalization of the inverse

propagator by the definition of renormalized Fermi level and interaction, EF andQ = (wR o/Ep )sc, respectively, in order to cancel the single dimensional pole, leads to the sameWilson function P(u) as for the compact nonlinear cr model when the scale parameter ir is varied atconstant Wo/Eo. The conductivity is calculated in a perturbation expansion in (~EF), with

'=mR'OE+ ' being the inverse lifetime, at d =2+@. It is explicitly shown that in ultraviolet-divergent d-dimensional loop integrals over advanced and retarded propagators, the leading term isregular while the dimensional pole occurs in the next-to-leading term. Then to leading order theconductivity o.o(co) is regular while the first correction that includes the diffusion modes has a di-mensional pole with residue =(im/~ )' . To cancel this pole a renormalized inverse conductancet(u) is defined, and the new Wilson function P(t) obtained by varying x at constant "bare" interac-tion 8'o/EF coincides with the scaling theory of Abrahams, Anderson, Licciardello, and Ramak-rishnan. Scaling laws are derived from the solution of the renormalization-group equation for theconductivity.

I. INTRODUCTION

The purpose of the present work is to study the locali-zation transition within a field theory where the generat-ing functional Z is the grand partition function for elec-trons in random potentials, or at T=O the vacuum-to-vacuum amplitude, ' by means of renormalized perturba-tion theory. Z is expressed as a functional integral overGrassman (anticommuting) fields.

The scaling theory for the localization transition ofelectrons in random potentials near and above two dimen-sions is now firmly established since Abrahams et al.showed that the dimensionless conductance g at T=O is auniversal function of the length scale of the systein. Fol-lowing earlier work by Thouless and Licciardello, theyobtained asymptotic forms for the scaling function P(g) tosecond order in the expansion parameter g

' for dimen-sionality d & 2. Along a different line of approachWegner first derived scaling laws for the conductivity,while in a series of following papers Wegner andSchaffer ' developed a Lagrangian formulation based onan effective functional that generates the configurationalaveraged product of two-particle Green's functions(G(zi)G(z2~), „, zi 2 being complex energies. Here theelectrons are described by commuting fields and it em-erges from this work that there would be two generatingfunctionals: One for the case sgn(Imzi) =sgn(Imz2) that ismapped into a compact generalized nonlinear o model,and a different functional for sgn(Irnzi)= —sgn(lmzz)that maps into a noncompact generalized nonlinear o.models with n =0 components because of the replicamethod. The difference between the two models is cru-

cial: The compact o. model does not have a fixed pointfor positive physical values of the reduced "temperature"t=g, while the noncompact model exhibits a positivefixed point t~ that describes the mobility edge in an ex-pansion in e=d —2. The conductivity belongs to the lastclass and the results for the noncompact nonlinear irmodel agree with the scaling theory of Abrahams et al.A detailed comparison of both models was given byHoughton et al.

Hikami derived an effective Hamiltonian for diffusionmodes that also maps into the noncompact nonlinear omodel, and he performed a three-loop order calculation inthe e expansion. His results show that the second andthird loop corrections identically vanish in the replica lim-it n =0, hence confirming the results of Ref. 2 up toO(e').

Efetov et al. were the first to introduce Grassmanvariables for the electron fields. Also in their work thegenerating functional is defined to reproduce correctly(G(zi)G(z2) )« for - sgn(Imzi )= —sgn(Imz2), and itmaps into the noncompact nonlinear o. model. Along dif-ferent lines, a self-consistent theory of Anderson localiza-tions was recently developed by Vollhardt and Wolfle. '

The purpose of this paper is twofold. The first purposeis to present a theory based on the fact that for a nonran-dom system the vacuum amplitude Z is the generatingfunctional of all correlation functions at T=O. This is notnew, and it has been applied before to the theory of super-conductivity" and the Kondo problem' ' among others.For a randoin system (Z")«=Z„ is the generating func-tional of the configurational averaged correlation func-tions in the n=O limit. The second purpose is to apply

28 6453 1983 The American Physical Society

Page 2: Second-quantized theory of Anderson localization in

ALBA THEUMANN

for the first time renormalized perturbation-theorymethods' and dimensional regularization' to a Lagrang-ian theory with unperturbed propagators that have a fin-ite, negative "square mass" —E0 and a cut in the complexfrequency plane, where ED is the Fermi energy. It shouldbe pointed out that for nonrelativistic electrons themomentum k and co do not form a four-vector and havedifferent dimensions, thus presenting a different problemthan in the relativistic theory of quantum fields.

In Sec. II the replica method is used to express Z„as afunctional integral over complex Grassman fields P (k, co)

that are frequency and momentum dependent, and thereplica index +=1, . . . , n. It is explicitly shown for corn-pleteness that the configurational averaged correlationfunctions are obtained through functional differentiationof Z„/n in the n=O limit. The singular behavior of thevertex functions is analyzed following standard field-theory methods.

The Lagrangian is that of a P(") theory with the fre-quency as an extra variable, and for this particular case ofrandom-scattering centers, the critical space dimensionali-ty remains d, =4. At d =2+e the theory is super-renormalizable and only two graphs have primitive ultra-violet divergences: The first-order contribution to theself-energy and the first diagram or leading conductivityo0(co) in Fig. 2(a). However, the explicit calculation in theAppendix shows that an ultraviolet divergent integral overretarded and advanced propagators, in an expansion in(~Ep ), ~ is the lifetime and EF the renormalized Fermienergy, has a regular leading term while the correspondingdimensional pole occurs in the next-to-leading term. Thento leading order cr0(co) is regular and equals

~(2 d)[l (y/2)] ( (1 g/2) Ed/2

There are two bare parameters in the theory: The Fer-mi level ED and the variance of the random potential 8'0that acts as a quartic coupling. It is shown in Sec. II thatthe inverse propagator is correctly renormalized by defin-ing EF——EDZ (u), with the dimensionless interaction

u =(mWp/EF. )a" =D.() ',DD being the diffusion constant and a the scale parameter.By varying ~ at constant dimensional 8'0/E0 one obtainsthe Wilson function P(u) that coincides with the P func-tion of the compact nonlinear o. model when n=0, whichdoes not have an infrared unstable fixed point. In Sec. IIIthe singular behavior of the conductivity as given in Fig.2(a) is analyzed, and it is shown that the localization tran-sition cannot be described within the canonical applicationof renormalized perturbation-theory methods. As wasmentioned, the only diagram with a primitive divergenceis o.0, and the removal of the dimensional pole by a renor-malization leads to the same result as in Sec. II. When~=0 the second contribution to the d —c conductivityalso has a dimensional pole that reflects the infraredsingularity. However, the residue of this pole also has the"wrong sign" in the sense that it combines with o.0 to givethe P function of the compact nonlinear o. model after re-normalization. An alternative renormalization procedureis presented also in this section, and it consists of keeping

only the leading terms in an expansion in 1/v.EF. Al-though all the integrals entering the calculation are ultra-violet convergent, it is found again to leading order a di-mensional pole with residue (ice/a)' . due to the small-ness of a next-to-leading singular term O(1/~Eq). The"bare" expansion parameter is now u =(rEF) and thedimensional pole is canceled by the definition of a renor-malized inverse conductance t = t (u). It is shown that byvarying ~ at constant bare W0/E~ one recovers the resultsof Abrahams et a/. The renormalization-group equationobtained to this order for the conductivity and scaling re-lations are also discussed in this section.

II. GENERATING FUNCTIONALANl3 RENORMALIZATION

( V(q )V(q ')),„=&05(q+q '), (2.2)

while P ( k ) and P( k ) are creation and destruction opera-tors, respectively.

At T=0 the vacuum-to-vacuum transition amplitudefor an infinite time interval can be expressed as a func-tional integral' '

Z = f&(Pt,P)e", (2.3)

with the action

A = fdk den(k E()—co)$ —(k,co)P(k, co)

+ —,' fdk dq der V(q )[Pt(k+ q, co)P(k, co)+H.c.]

+ fdq de j (q, co).A@M(q, co), (2.4)

and where the last term includes the coupling of thecurrent density,

j (q, co)=2e fdkdQ k$ k+ —;0+—2' 2

yP k ——,0——q 6)2' 2

(2.5)

to the vector potential AzM(q, co). The integrals over thefrequencies run from —oo to + ao. Here P(k, co) is acomplex, anticommuting e variable that can be expressedin terms of two real, independent Grassman variables:

P(k, co) = [g'(k, co)+ig(k, co)],1

(2.6)

P (k, co) = [g(k, co) —iq(k, co)],2

The Hamiltonian for a system of spinless electrons inthe presence of random potentials in d dimensions is

H —EQN= fdk(k ED)pt—(k )p(k )

+ —,' fdk dq V(q )[P (k+q )P(k )+H.c.],

(2.1)where the electron mass m = —, for simplicity, EQ is the

Fermi energy, and a differential dk includes a factor(2m )

The probability distribution of V(q ) is assumed to beGaussian with zero mean and with variance

Page 3: Second-quantized theory of Anderson localization in

SECOND qUANTIZED THEORY OF ANDERSON LOCALIZATION. . . 6455

that satisfy

Ig, 'riJ =If,g'I=['i)', i1I =0, g ='rI =0,fdg= fdg=0, fding= fdye= 1 .

It follows from Eqs. (2.6) and (2.7):

$2=(pt) =0, ptp=igri=

(2.7)

(2.8)

If one introduces the auxiliary fields

A —+A+ fdk den fdk 'de' v[k, co;k 'ro']Pt(k, ro)P(k ', co')

(2.9)

in Eq. (2.3), the p particles Green's function is formallygiven by

55u (1,2)

=(P (1)P(2) P (p)P(p+1)) (2.10)lnZ5u(p, p + 1)Gp(1, 2, . . . , p,p+1)=

IuI =0

Here ( ) indicates an expectation value, while ( ),„indicates a configurational average.The configurational average over the random potential is obtained from Eq. (2.10) either by expanding the right-hand

side (rhs) in terms of V(q ) and averaging term by term, which reproduces the known perturbation expansion, or by us-

ing from the start the replica method. ' With the use of the latter approach I obtain from Eqs. (2.2)—(2.4) and (2.10)

((y'(1)y(2) ~ ~ ~ y'(p)y(p+1) ».,=1 5

where

(zN) z f+~yty e 0 1

—(z &.„5v(p,p+1) n

(2.11)

(2.12)

Ao ——fd k dc@(k Eo co—i g—sgn—co)g P~( k, co)P~( k,co),a

AI= &Of dkdk'dq fdcodco'gp (k+q, co)p (k,co}p&(k' q, co')QIi(—k', co')+ fdq dcoAzM(q, co)g j (q, co) .

(2.13)

a,P

(2.14)

The replica indices a,P run from 1 to n, and the infinitesimal iinaginary part was added to the inverse propagator in Eq.(2.15) to obtain a correct expansion in terms of causal Green's functions.

Linear-response theory gives for the configurational averaged expectation value of the current density, from Eqs.(2.11)—(2.14),

( (j„(q, co ) ) ),„=lim —g QAz~~ (j„(q, co )j~&( —q, —co) )n~On p

(2.15)

where the expectation value on the right-hand side (rhs) is

(( )&= ' f~up'. y.e'""""( ~ ) .n a

(2.16)

The subscripts x,y indicate the space components of thevectors j and Az~. In the limit n=0 only the connectedpart of the current-current correlation function will con-tribute; it has a=P and the sum gives a factor n. Alldisconnected diagrams will vanish in the n=O limit be-cause they include an independent sum over a and P, thusan extra factor of n in the numerator. The expression forthe conductivity tensor is then recovered,

cr y(q, co}=lim —g (j (q, co)j y( —q, —co))1 . . 1

n —+0 n CO

(2.17)

with j ~(q, co) as in Eq. (2.5), and where ( ), indicates asum over connected diagrams in a perturbation expansionin terms of Wo. The vector potential AEM was set equalto zero.

The Lagrangian theory defined by the generating func-tional of Eqs. (2.12)—(2.14) differs in one iinportant as-pect from the theory of Efetov et al. While in the

present formalism all correlation functions can be ob-tained from Eq. (2.11), and the imaginary part i g sgn(co) isdictated by causality, the choice of complex energies zz inRef. 9 seems arbitrarily dictated by the requirement of de-fining a generating functional for (G„Gz ),„,where G~~g~are the advanced and retarded propagators. The same ap-plies to the functionals of Refs. 4 and 5.

All correlation functions can be expanded in a diagram-matic series in powers of IVO, with the unperturbed in-verse propagator

I o '(k, co) =Go '(k, co) =k —Eo —co i risgnco, —(2.18)

(P~( k co)fp( k co ) )0=5~@5(k —k )5(co —co )Go( k co)

and with the four-point interaction of Fig. 1(a). The anti-commutation of the Grassman variables automaticallygives Wick's theorem. A closed fermion line carries a fac-tor n and it vanishes when n=O. As the interactiontransfers momentum but no frequency, the only nonvan-ishing diagrams have open lines and constant frequencyalong the lines. The full propagator G(k, co) is shown in

Fig. 1(b) and the self-energy X(k, co) is shown in Fig. 1(c).The conductivity is shown in Fig. 2(a} in terms of thefour-point vertex function I' '. The "diffusion propaga-

Page 4: Second-quantized theory of Anderson localization in

6456 ALBA THEUMANN 28

K)4)

K )(d

(oo(q, M j

lq Q (d

X X

(dK- —q, Q-—2 '

2

K+& q K'+-,'gQ+U)

1K- —$2

0-—(dK-2q

+

(c)+ o ~ ~

FICz. 1. (a) Effective electron-electron interaction. o.,P= l, . . , n are .replica indices. (b) Full propagator and (c) firstcontributions to the self-energy when n =O.

+ 0 ~ ~

FIG. 2. (a) Full expression for the conductivity o.( q, co). I' ' isthe complete vertex function and every cross carries a factor k.(b) Partial summation of multiply crossed diagrams for I D'.

tor" I D' is the contribution to I' ' of the partial summa-tion of multiple cross diagrams in Fig. 2(b), and when in-serted in Fig. 2(a) gives the singular behavior of the con-ductivity in two dimensions. '

The great advantage of having a Lagrangian theorywith a well-defined generating functional is that the diver-gences that occur in a perturbation expansion in d dimen-sions can be treated systematically by using renormalizedperturbation-theory methods. ' ' Dimensional analysisapplied to the action in Eqs. (2.13) and (2.14) shows that[Wo] =A "for A an inverse length.

Then the critical dimensionality of the problem isd, =4, and for d =2+@ the theory is super-renormalizable. This means that only a few diagrams willhave primitive ultraviolet divergences that can be canceledby the introduction of renormalized parameters in place ofthe bare, physical parameters. ' ' The degree of diver-gence of a diagram y with r vertices in the expansion of avertex function with E external legs is, as in the usual P' '

theory, ' '

5 (y)=5&(y) 8+2+—2d =r(d —4)+(d —2), (2.20)

where 54=(r —1)(d —4) from Eq. (2.19). For the conduc-tivity the only ultraviolet divergence is for r=0, or thefirst diagram of Fig. 2(a).

By using the method of dimensional regularization, ' 'the integrals are performed at d & 2 when they are conver-gent and are analytically continued to d =2+@. The loga-rithmic singularities will appear now as dimensional polesin e that are removed by the introduction of renormalizedparameters. The method used to perform loop integrals ind dimensions when the propagators have complex poles isdiscussed in the Appendix. The only singular contribu-tion to the inverse propagator is then

(2.21)

where

Xt(ro) = Wc2 I dk k 'Gc(k, ro)

5~(y) =d —,' E(d —2) +r (d ——4), (2.19)= Wc(En+co)' —)r cot +i)r sgn(ro)

2

because the open lines carry constant frequency and theloop integrations are only over "internal" momentumvariables in the limit n=O. One thus concludes from Eq.(2.19) that all contributions to the N-point vertex functionI' ' are regular for N & 2, while there is only one diver-gent diagram with r =1 in the expansion of I' ' for2&d &3. This is the lowest-order contribution to theself-energy shown in Fig. 1(c). The degree of divergenceof the conductivity is easily calculated from Fig. 2(a): Theconductivity is obtained by joining four propagators toI' ', one factor k at each cross and two loop integrations.Then

(2.22)

and a factor Kd ——2 )r ~ [I'(d/2)] ' was absorbed inWo. ReX&(co) is a constant, independent of k and co,

~

ro~

&&Ec, and from Eqs. (2.18), (2.21), and (2.22) one ob-tains the singular contribution to the Fermi energy,

e/2Ei ——Ep ——8'pE p (2.23)

There are no poles of higher order in e. In Refs. 9 and 19ReX~ was set equal to zero.

In order to remove the dimensional pole the bare, physi-

Page 5: Second-quantized theory of Anderson localization in

SECOND-QUANTIZED THEORY OF ANDERSON LOCALIZATION. . . 6457

cal parameters Eo, W'0 are expressed in terms of a renor-malized EF and a dimensionless interaction u,

Ep =EFZ, (2.24)

8'0 ———EFu K2—d (2.25)

with K a scale parameter,

(2.26)

and no term O(u ) on the rhs of Eq. (2.26). From Eqs.(2.21)—(2.25) the renormalized inverse propagator is, tofirst order in 8'0,

I'x (k, co) =k EF co—i —sgn—co,—(2) 2 . 1

rwith

(2.27)

e/2—=~8'()Ep ——EFu . (2.28)

The higher-order contributions to X do not have ultra-violet divergences and are negligibly small compared to X&

because the loop integrals are over propagators with thesame frequency sign, as discussed in Eqs. (A8) and (A9).Equation (2.28) shows that the expansion parameter in thetheory is

1u~0

(2.29)

where Do rE~ is the ——diffusion constant, in agreementwith Ref. 8.

In terms of the bare dimensionless coupling

III. SINGULAR BEHAVIOROF THE CONDUCTIVITY

I D'(k, k ', co) = 8'0 rEFIk+k'I'

r d(3.2)

to leading order in I /~EF. A factor

Kd ——2 n [I (d /2) ]

The scaling theory of Abrahams et al. is based on aperturbation expansion in terms of the inverse conduc-tance. Recently it has been pointed out by Castellaniet al. ' that this result follows from a gauge-invarianttheory and the multiplicative renormalization group.Hence it is relevant to discuss here the conductivity as it isobtained from Eq. (2.17) within the framework of renor-malized perturbation theory and dimensional regulariza-tion. All the integrals entering the calculation of the con-ductivity are then performed at general dimensionality d,while all published results ' are for d=2. The loop in-tegrals over propagators with complex energy polespresent different features than the integrals appearing inthe theory of critical phenomena' and they are discussedin the Appendix. The order of magnitude of relevantphysical quantities is

&&EF ~

I(3.1)

The conductivity is obtained from Fig. 2(a) with the in-verse propagators I Pf' of Eq. (2.27) and the partial sum ofmaxiinally crossed diagrams I D' of Fig. 2(b) for the four-point vertex function. I D' is expressed in terms of con-vergent integrals and is calculated in the Appendix forcompleteness. Gne obtains the known result

r

uo n( Wo/Eo)K-—Equations (2.24)—(2.26) give

up

21 — up

ATE'

The Wilson p function is given by

(2.30)

(2.31)8pd

o(co) =4e Jo+ J&(co)EF

(3.3)

where

was absorbed in 8'p.By keeping only the integrals over propagators with op-

posite frequency sign one obtains for the conductivity

p(u)= K ua

BK=eu+ —u

2 2

W'0/E((2.32) JO +d2

k —E+ k —E(3 4)

where the scale parameter K is varied, keeping constant thephysical parameters Ep 8 p. This is an exact result andthere are no correction terms O(u ), I & 3, on the rhs ofEq. (2.33). The absence of correction terms was proved byHikami to 0 (u ) by the mapping to the generalized non-linear o model. The P function obtained in Eq. (2.33) isthe same as for the compact nonlinear o model ' ' withn =0 components, or for the generalized nonlinear cr

model with symplectic symmetry, and it does not have aninfrared unstable fixed point. The equation P(u~) =0 doesnot accept a nontrivial solution u* & 0.

This result originates in the renormalization of thetheory, keeping constant the bare parameters S0 Ep.However, the scaling theory of localization follows froman expansion in terms of the inverse conductance 'u = I /~EF, and it is discussed in Sec. III.

COdCO=

rEF

E+ EF+i /~ . ——

(3.6)

(3.7)

The zeroth-order contribution Jo in Eq. (3.4) is logarith-mically divergent at d=2, while Ji(co) in Eq. (3.5) is con-vergent for d & 4, and it is free of infrared divergences form &0, in agreement with the power counting analysis Eq.(2.20).

One obtains from Eq. (A8)

Jt, (co)=I' ' f fdkdk'Ik+k'I2 iso k E+——

x . . . , (3.5)1 1 1

k —E k' —E+ k' —E

Page 6: Second-quantized theory of Anderson localization in

ALBA THEUMANN 28

Jp ——Kd m.~EF 1 — —cot — +01 d 77d

REF 2 2

(3.8)

One observes that the leading term in Jp is regular,while the dimensional pole is in the next-to-leading term.The evaluation of Ji (cp) in d =2+e dimensions isdescribed in detail in the Appendix. From Eqs. (A13) and(A18) one obtains

2d dn 1

[ 1(~)]sing=Kd yy 2 E 20 (y E+ y—E—()g)ei2'rEdn

EF 2

EF 2 Wp1+— (i cp)~Wp e EF

(3.14)

where r '=m. WpEg '. This equation for cT(cp) shouldbe compared with Castellani et al. ,

' where the dimen-sional pole is replaced by an arbitrary cutoff A at d =2.In terms of the dimensionless coupling u of Eq. (2.25),

e/2a(cp), 1 21+e Q 7T'6

QK

(3.15)

particular, we cannot tell at the present stage if otherphysical quantities will have primitive divergences that re-quire further renormalization. This new approach doesnot invalidate the results of Sec. II. There the first contri-bution to self-energy Xi was evaluated in full in Eq. (2.22),and the imaginary part just defines r

If one follows this approach, it is obtained by introduc-ing Eqs. (3.12) and (3.13) in Eq. (3.3) with K2 (4n——)

r

where now u appears as the bare coupling. To cancel thedimensional pole in Eq. (3.15) one introduces a renormal-ized coupling,

Only the terms with a dimensional pole and the in(cp)singularity were kept in Eq. (3.9). At d =2, Eq. (3.9) givesto leading order, from Eq. (A20},

r

Q

1+2/~au

and one obtains

(3.16)

[Ji(cp)]„„=Kd rEF ln—2 EF

(3.10)o(cp), 1 t1+—ln

ldll

K V.EF2 (3.17)

where the ellipsis signifies the regular terms.Introducing Eq. (3.10) and only the leading term of Eq.

(3.8) in Eq. (3.3), one obtains the result of Gor'Kovet al. Hence at finite cp the theory is free of divergences,except for the next-to-leading contribution to crp in Eq.(3.8) that is usually neglected. The dc conductivity has in-stead a dimensional pole that reflects the infrared diver-gence when co=0. One obtains from Eqs. (3.3), (3.8), and(3.9)

o.(0)=err' 1 —— (3.11)

A quick comparison with Eq. (2.23) shows that the resi-due of the pole has again the wrong sign. Renormaliza-tion of Eq. (3.11) will lead to a p function as in Eq. (2.32)that does not have an infrared unstable fixed point.

As an alternative one may conjecture that only the lead-ing terms in 1/~EF should be kept in all integrals; thenfrom Eq. (3.8) one obtains

Jp ——EdREF 1+0d/2 1

~EF(3.12)

and in Eq. (3.9) the last term inside the large squarebrackets can be neglected to yield

e/2[Ji(co)]„„g Kdirr EF—— (3.13)

It is obvious that in this way we are violating the powercounting analysis of Sec. II, and that we are not doingstrict renormalized perturbation theory to all orders. In

where t is the renormalized inverse conductance. The crit-ical behavior is determined by a new P function,

p(t) =Ic atBK

(3.18)

Wo /EFwhich is the result first predicted by Abrahams et al.The localization transition is governed by the infrared un-stable fixed point t*=em. l2. The absence of higher-orderterms was proved by Hikami within the mapping to thenoncompact nonlinear o model, although it is difficult toprove in this context. In terms of the renormalized cou-pling t of Eq. (3.16) it is obtained for Ep and Wp in Eqs.(2.24) and (2.25)

1 20=—Eo

2Ep ——EF 1

&E' (3.19)

oui ( Wp,'Ep, cp) =o g (t;E~,'lc",cp), (3.20)

and the renormalization-group (RCx} differential equationfor o.z follows only from the condition that o.z is indepen-dent of K. It is obtained to this order in the expansion:

K +P(t) +B(t)E~ cr~(t;E~, lc;co) =0& (3.21)

8BK Bt REF

where

The relationship between the bare and renormalizedconductivity ' is

Page 7: Second-quantized theory of Anderson localization in

SECOND-QUANTIZED THEORY OF ANDERSON LOCALIZATION. . .

e(t) =a lnEFa

BK

1 P—(t) —e . (3.22)

Wo /EF' Eo

The RG equation in Eq. (3.21) with B(t) and P(t) asgiven in Eqs. (3.22) and (3.18) is analogous to the equationof the compact nonlinear o. model' for n &2. In thepresent theory there is no wave-function renormalization,as is the case in Ref. 8 for n =0. Also it is relevant topoint out that the role of the field H in the o model is tak-en here by EF, and not by the external variable cu as it isusually assumed. '

The differential Eq. (3.21) can be integrated' to give—1 '1/e"

o g(co;t, EF,a)=4 co;EF 1 ——t jfc

~Kt t*

(3.23)

where N(co;x;y) is an arbitrary function of two indepen-dent variables x,y, and t & t*. In terms of the correlationlength'8

1=K—1 (3.24)

one derives from Eq. (3.23) the scaling relation for a = 1

oz(co, t,EF)='g ' 'N cog;EFg ;1-t' (3.25)

where d —2 is the canonical dimension of o.. From Eq.(3.19) it is natural to scale

t*—t=W, —W, (3.26)

(=i', —wi ", v=—1

E(3.27)

and the scaling relation for the dc conductivity (co=0)

&dc =EF f(d —2) /d

w, —w i'

where W is the width in the distribution of random poten-tials. ' It is not clear in this formulation how to introducethe mobility edge E, . From Eqs. (3.24) and (3.26) followsthe known result

functional integral over Grassman variables, thus makingthe introduction of effective functionals unnecessary aswas done in previous work. The second quantized elec-tron fields P (k, co) [P(k,co) j represent the Fouriertransform of the Heisenberg creation (destruction) opera-tors P (k, t) [P(k, t)] (t) being the real time and to the fre-quency. " Through the use of the replica method atranslational invariant generating functional is obtainedfor all the configurational averaged correlation functionsin the limit n =0. The physical parameters of the theoryare the Fermi energy Eo and the variance of the randompotential Wo, which acts as a quartic coupling. It wasfound that despite the integration over the frequencies thecritical dimensionality of the system is d, =4; therefore,for d =2+a only a few diagrams have ultraviolet diver-gences that are removed by the introduction of renormal-ized parameters.

These are the first contribution to the electron self-energy and the zeroth-order term in the conductivity. Thesingularity in the self-energy leads to a "mass renormali-zation" term or to a renormalized Fermi energy EF. Itwas discussed in Sec. II that the renormalization of thetheory, keeping constant the ratio of bare physical param-eters Wo/Eo, leads to the Wilson P function of the gen-eralized nonlinear o. model with symplectic symmetryand n =0 components that does not have an infrared un-stable fixed point.

To describe the localization transition it is essential toanalyze the singular behavior of the conductivity as afunction of the dimensionality d, as is discussed in Sec.III. Power counting shows that the only diagram with alogarithmic divergence is the first diagram of Fig. 2(a).However, an explicit calculation shows that the leadingterm in an expansion in (rEF) ' is regular while the di-mensional pole occurs in the next order. This is a salientfeature of this theory that has not been encountered beforeand merits further discussion. When an ultraviolet diver-gent integral is over propagators with complex poles atopposite sides of the branch cut in Fig. (3), the phasedifference in the asymptotic region EF »~ ' cancels thedimensional pole in the leading term.

The results of Sec. III show that the localization transi-tion cannot be described within the strict framework of re-

(3.28)

Equation (3.28) was derived by Hikami for o(co) withco in place of EF. In his formulation the frequency co isinterpreted as the analogous term to the magnetic field Hfor a spin system, while here cu is always an external vari-able; it is not a parameter of the theory.

IV. CONCLUSIONS

The localization transition of electrons in random po-tentials was analyzed by means of renormalized perturba-tion theory and dimensional regularization. It was point-ed out that the generating functional of the system atT=O is the vacuum-to-vacuum amplitude' expressed as a

/FICx. 3. Contour C of integration in Eq. (A3). The phases at

either side of the branch cut are explicitly shown.

Page 8: Second-quantized theory of Anderson localization in

ALBA THEUMANN 28

normalized perturbation theory. As discussed before, forco&0 only the zeroth-order contribution 0-0 is divergent,and the removal of the dimensional pole by renormaliza-tion will again lead to a p function that does not have aninfrared unstable fixed point. The same result follows ifone looks at o.d, . In this case the next-order contributionalso has a pole, indicating the infrared divergence at d =2.Unfortunately, the residue of this pole also has the wrongsign and it leads to the "wrong" p function, as was dis-cussed after Eq. (3.11). To find a way out of this impasse,it was conjectured that all relevant integrals should beevaluated only to leading order in 1/ATE+.

One then has to face the unusual situation that to lead-ing order in (rEF) an integral with a primitive diver-gence is regular, like Jp in Eq. (3.8), while a convergent in-tegral like Ji(co) in Eq. (3.9) has a dimensional pole [Eq.(3.13)] to leading order. In order to remove this pole afurther renormalization of the conductivity in Eq. (3.14) isneeded, and this leads to the definition of a renormalizedinverse conductance t in Eq. (3.17). Here the bare dimen-sional parameter is 8'p/E~. The known results for thescaling theory of the localization transition ' follow, in-cluding a scaling relation that was obtained before fromthe mapping to the noncompact nonlinear 0. model.

It is obvious that these results do not follow from acanonical use of renormalized perturbation theory. Powercounting is violated; therefore, we cannot predict at thepresent stage the degree of divergence of other diagrams,or if there exist other physical quantities that need renor-malization.

ACKNOWLEDGMENTS

By calling C the contour indicated in Fig. 3 one obtains

In, m (1 i2nP) —1 dZZP 1(Z E ) n(Z E )—m

C

=(1 e—' ~&) '2mi(R. n+Rm ), (A3)

where the branch cut of z" ' is along the positive real axisand Rn, R~ are the residues at the poles,

gn —1

z~-'(z —E,.)-n( 1)1 ~ n 1 s z=E

S

(A4)

and the analogous equation for Rm, for the values of pwhere the integral converges. The interest here lies in thevalue of the integrals for

1&&EF p (AS)

in an expansion in 1/rE+. From Eqs. (A2), (AS), and thephases as indicated in Fig. 3,

E E i(2m' —Q)Fe 7 (A6)

y=tan-' 0(it (n-.F

Let us evaluate first the integral I'+' for the leadingcontribution to the conductivity in Eq. (3.3). From Eqs.(A2)—(AS) with iM =d/2+ 1,

( 1 eind) —1 [('E )d/2 (E )d/2]

I thank the Aspen Center for Physics during the AspenWorkshop on Localization for their hospitality. I ack-nowledge stimulating discussions with Professor A.Houghton, Dr. S. Hikami, and Dr. W. K. Theumann. Iam grateful to Professor E. Abrahams for a very clarify-ing discussion that led to the ideas presented in this paper.Part of this work was done at the Department of Physicsof the University of Alabama. We thank the Brazilianagencies Conselho Nacional de Pesquisas (CNPq) andFinanciadora de Estados e Projetos (FINEP) at theUniversidade Federal do Rio Grande do Sul for their sup-port.

APPENDIX

The d-dimensional one-loop integrals occurring in thecalculation of the conductivity are of the general form

and from Eq. (A6),

I+ ——m~EF 1 — —cot m —+01 d d

F

1

REF

=mrE d(1 e' ) '(~E ) '—+0

(AS)

The leading term is regular when d =2, while the resi-due of the dimensional pole is O(l/rE+). In the samefashion it is obtained for I++, with the poles at the sameside of the cut,12,P 2

~

( 1 aid) —1 (E )d/2 —1

+

In, m d( E)n ( E )m

with

(Al)(A9)

This integral has indeed a dimensional pole but the con-tribution is O(1/rEF) compared with I'+' . I D'(k, k ', co)in Fig. 2(b) is given by

lEs EF+s co+—

s=+ or—

(A2) I D (k, k', co)= JYp[1 —Wpy(k+k ', co)]

with

(A10)

(A11)y(q, pi)=+d fdk = f dx f z dzi k+q i

2 E p p [z+x(1 x)q' (1 x)E+ x—E ]—— —

Page 9: Second-quantized theory of Anderson localization in

SECOND-QUANTIZED THEORY OF ANDERSON LOCALIZATION. . . 6461

By expanding in q one obtains to leading order

y( q, co) =m.rEF2rEF

1 —EN7~ g

2 (A12)

and by introducing Eq. (A12) in Eq. (A10) together with Eq. (2.28) one arrives at Eq. (3.7).In order to calculate J& (co) in Eq. (3.4) it is convenient to write

1Ji(co)= fdk 2, Hi(k, co),k' —E k' —Ewhere, with the use of Eq. (A3),

(A13)

and

H, (k,co)=Kd ' fdk '

~k+ k '

~

2 —ice k' E+—= —k, .d

——1 [N+(k, co) N(k,—co)]E E 1 m (A14)

1

N+(k, co)= f dxx[E+(1 x)+i—cox —x(1 —x)k ] ~

To arrive at Eq. (A14) one uses the identity'

=f dx[~

k '+xk~

+x(1 x)k —i~x —E(1—x—)]~k+k

~

—leo k E+

(A15)

The integral N+(k, co) in Eq. (A15) is logarithmically divergent when co=0 and d =2. To isolate the singularity in cofor d =2+a one introduces the identity

[2xk —(E+ ico+k )—+(1 x)(k +E+—iso)]-k —E+ +iso

in Eq. (A15), thereby obtaining

N (k ~)= [(i~) ' ' —(E )' ']

d —2 k —E++im

+ f dx(l x)[E+(1 x—)+icox——x(1—x)k ]k +E+ —1 co

2 d/2 —2

k —E+ + l co

(A16)

(A17)

In Eq. (A17) the first term contains the dimensional pole and the ln(co) singularity when d =2. The second integral isregular for co =0 and d =2, and it can be ignored.

From Eqs. (A17), (A14), and (A6)

[Hl(k ~)]si.s=, , —(i~)'" «F ——+k 1 2 . zg2 dy2 (EF k ) e' 1

k —E+ k —E e EF 2 &EF(A18)

The leading term in the large parentheses of Eq. (A18) is proportional to e, while the next-to-leading term is not. Ifone neglects terms 0(1/~EF ), one obtains from Eqs. (A18) and (A13)

[J& (co)]si„s K~C (i co)'~—— —E'

where

(A19)

C= f dyy~" 1 1 m 3 d/2=—z EF 1+0 1

(y E+ )' (y E—)' 2 —«F (A20)

The last equality follows from Eqs. (A3), (A4), and (A6).

S. F. Edwards, J. Phys. C 8, 1660 {1975).2E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V.

Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).3D. C. Licciardello and D. J. Thouless, Phys. Rev. Lett. 35,

1475 (1975).4F. J. Wegner, Z. Phys. B 25, 327 (1976).5F. J. Wegner, Z. Phys. B 35, 207 {1979).6L. Sehafer and F.J. Wegner, Z. Phys. B 38, 113 {1980).

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